metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).45D12, (C2×C12).56D4, (C22×S3).5Q8, C22.51(S3×Q8), C6.63(C4⋊D4), C2.9(D6⋊3Q8), (C2×Dic3).62D4, C22.250(S3×D4), (C22×C4).122D6, C6.51(C22⋊Q8), C2.26(C12⋊D4), C2.11(C12⋊7D4), C3⋊3(C23.Q8), C23.32D6⋊21C2, C2.17(C4.D12), C2.23(D6⋊Q8), C22.130(C2×D12), C6.30(C42⋊2C2), (S3×C23).23C22, (C22×C6).358C23, C23.393(C22×S3), (C22×C12).69C22, C2.13(C23.14D6), C22.111(C4○D12), C22.54(Q8⋊3S3), C22.105(D4⋊2S3), (C22×Dic3).63C22, (C6×C4⋊C4)⋊23C2, (C2×C4⋊C4)⋊12S3, (C2×C6).86(C2×Q8), (C2×D6⋊C4).24C2, (C2×C4⋊Dic3)⋊14C2, (C2×C6).338(C2×D4), (C2×Dic3⋊C4)⋊42C2, (C2×C6).87(C4○D4), (C2×C4).44(C3⋊D4), C2.15(C4⋊C4⋊S3), C22.143(C2×C3⋊D4), SmallGroup(192,553)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).45D12
G = < a,b,c,d | a2=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b2c-1 >
Subgroups: 552 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×9], C22 [×7], C22 [×10], S3 [×2], C6 [×7], C2×C4 [×4], C2×C4 [×17], C23, C23 [×8], Dic3 [×4], C12 [×5], D6 [×10], C2×C6 [×7], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C24, C2×Dic3 [×2], C2×Dic3 [×8], C2×C12 [×4], C2×C12 [×7], C22×S3 [×2], C22×S3 [×6], C22×C6, C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×6], C3×C4⋊C4 [×2], C22×Dic3 [×3], C22×C12 [×3], S3×C23, C23.Q8, C23.32D6, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4 [×3], C6×C4⋊C4, (C2×C4).45D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×2], C23, D6 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4 [×3], C22⋊Q8 [×3], C42⋊2C2, C2×D12, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C23.Q8, C12⋊D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C12⋊7D4, C23.14D6, D6⋊3Q8, (C2×C4).45D12
►On 96 points
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 35)(14 36)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 85)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 73)(71 74)(72 75)
(1 83 15 91)(2 92 16 84)(3 73 17 93)(4 94 18 74)(5 75 19 95)(6 96 20 76)(7 77 21 85)(8 86 22 78)(9 79 23 87)(10 88 24 80)(11 81 13 89)(12 90 14 82)(25 54 41 68)(26 69 42 55)(27 56 43 70)(28 71 44 57)(29 58 45 72)(30 61 46 59)(31 60 47 62)(32 63 48 49)(33 50 37 64)(34 65 38 51)(35 52 39 66)(36 67 40 53)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 14 15 12)(2 11 16 13)(3 24 17 10)(4 9 18 23)(5 22 19 8)(6 7 20 21)(25 40 41 36)(26 35 42 39)(27 38 43 34)(28 33 44 37)(29 48 45 32)(30 31 46 47)(49 95 63 75)(50 74 64 94)(51 93 65 73)(52 84 66 92)(53 91 67 83)(54 82 68 90)(55 89 69 81)(56 80 70 88)(57 87 71 79)(58 78 72 86)(59 85 61 77)(60 76 62 96)
G:=sub<Sym(96)| (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,85)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,73)(71,74)(72,75), (1,83,15,91)(2,92,16,84)(3,73,17,93)(4,94,18,74)(5,75,19,95)(6,96,20,76)(7,77,21,85)(8,86,22,78)(9,79,23,87)(10,88,24,80)(11,81,13,89)(12,90,14,82)(25,54,41,68)(26,69,42,55)(27,56,43,70)(28,71,44,57)(29,58,45,72)(30,61,46,59)(31,60,47,62)(32,63,48,49)(33,50,37,64)(34,65,38,51)(35,52,39,66)(36,67,40,53), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,14,15,12)(2,11,16,13)(3,24,17,10)(4,9,18,23)(5,22,19,8)(6,7,20,21)(25,40,41,36)(26,35,42,39)(27,38,43,34)(28,33,44,37)(29,48,45,32)(30,31,46,47)(49,95,63,75)(50,74,64,94)(51,93,65,73)(52,84,66,92)(53,91,67,83)(54,82,68,90)(55,89,69,81)(56,80,70,88)(57,87,71,79)(58,78,72,86)(59,85,61,77)(60,76,62,96)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,85)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,73)(71,74)(72,75), (1,83,15,91)(2,92,16,84)(3,73,17,93)(4,94,18,74)(5,75,19,95)(6,96,20,76)(7,77,21,85)(8,86,22,78)(9,79,23,87)(10,88,24,80)(11,81,13,89)(12,90,14,82)(25,54,41,68)(26,69,42,55)(27,56,43,70)(28,71,44,57)(29,58,45,72)(30,61,46,59)(31,60,47,62)(32,63,48,49)(33,50,37,64)(34,65,38,51)(35,52,39,66)(36,67,40,53), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,14,15,12)(2,11,16,13)(3,24,17,10)(4,9,18,23)(5,22,19,8)(6,7,20,21)(25,40,41,36)(26,35,42,39)(27,38,43,34)(28,33,44,37)(29,48,45,32)(30,31,46,47)(49,95,63,75)(50,74,64,94)(51,93,65,73)(52,84,66,92)(53,91,67,83)(54,82,68,90)(55,89,69,81)(56,80,70,88)(57,87,71,79)(58,78,72,86)(59,85,61,77)(60,76,62,96) );
G=PermutationGroup([(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,37),(10,38),(11,39),(12,40),(13,35),(14,36),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,85),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,73),(71,74),(72,75)], [(1,83,15,91),(2,92,16,84),(3,73,17,93),(4,94,18,74),(5,75,19,95),(6,96,20,76),(7,77,21,85),(8,86,22,78),(9,79,23,87),(10,88,24,80),(11,81,13,89),(12,90,14,82),(25,54,41,68),(26,69,42,55),(27,56,43,70),(28,71,44,57),(29,58,45,72),(30,61,46,59),(31,60,47,62),(32,63,48,49),(33,50,37,64),(34,65,38,51),(35,52,39,66),(36,67,40,53)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,14,15,12),(2,11,16,13),(3,24,17,10),(4,9,18,23),(5,22,19,8),(6,7,20,21),(25,40,41,36),(26,35,42,39),(27,38,43,34),(28,33,44,37),(29,48,45,32),(30,31,46,47),(49,95,63,75),(50,74,64,94),(51,93,65,73),(52,84,66,92),(53,91,67,83),(54,82,68,90),(55,89,69,81),(56,80,70,88),(57,87,71,79),(58,78,72,86),(59,85,61,77),(60,76,62,96)])
42 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 3 | 4A | ··· | 4F | 4G | ··· | 4L | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | Q8 | D6 | C4○D4 | D12 | C3⋊D4 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | (C2×C4).45D12 | C23.32D6 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×D6⋊C4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×S3 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 2 | 4 | 2 | 3 | 6 | 4 | 4 | 4 | 1 | 1 | 1 | 1 |
Matrix representation of (C2×C4).45D12 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[10,10,0,0,0,0,3,7,0,0,0,0,0,0,3,3,0,0,0,0,10,6,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[10,6,0,0,0,0,3,3,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,5,0,0,0,0,0,0,8] >;
(C2×C4).45D12 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{45}D_{12} % in TeX
G:=Group("(C2xC4).45D12"); // GroupNames label
G:=SmallGroup(192,553);
// by ID
G=gap.SmallGroup(192,553);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,184,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations